OFFSET
1,2
COMMENTS
The Fibonacci cube Gamma(n) is obtained from the n-cube Q(n) by removing all the vertices that contain two consecutive 1s.
The entries in row n are the coefficients of the Hosoya polynomial of the Fibonacci cube Gamma(n).
T(n,1) = A001629(n+1) = number of edges in Gamma(n).
Sum of entries in row n = A191797(n+2).
Sum(k*T(n,k), k>=1) = A238419(n) = the Wiener index of Gamma(n).
LINKS
S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
FORMULA
G.f.: tz/((1-z-z^2-tz-tz^2+tz^3)(1-z-z^2)). Derived from Theorem 4.1 of the Klavzar-Mollard reference in which the g.f. of the ordered Hosoya polynomials is given.
EXAMPLE
Row 2 is 2,1. Indeed, Gamma(2) is the path-tree P(3) having vertex-pair distances 1,1, and 2.
Triangle starts:
1;
2,1;
5,4,1;
10,11,6,1;
20,28,21,8,1;
MAPLE
g := t*z/((1-z-z^2-t*z-t*z^2+t*z^3)*(1-z-z^2)): gserz := simplify(series(g, z = 0, 20)): for j to 18 do H[j] := sort(coeff(gserz, z, j)) end do: for j to 13 do seq(coeff(H[j], t, k), k = 1 .. j) end do; # yields sequence in triangular form
MATHEMATICA
Rest /@ Rest[CoefficientList[CoefficientList[Series[t z/((1 - z - z^2 - t z - t z^2 + t z^3) (1 - z - z^2)), {z, 0, 10}, {t, 0, 5}], z], t]] // Flatten (* Eric W. Weisstein, Dec 11 2017 *)
DeleteCases[CoefficientList[Series[t z/((1 - z - z^2 - t z - t z^2 + t z^3) (1 - z - z^2)), {z, 0, 10}], {z, t}], 0, {2}] // Flatten (* Eric W. Weisstein, Dec 11 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 18 2014
STATUS
approved